Introduction to Game Theory, Behavior and Networks

Networked LifeCIS 112

Spring 2008Prof. Michael Kearns

Game Theory• A mathematical theory designed to model:

– how rational individuals should behave– when individual outcomes are determined by collective

behavior– strategic behavior

• Rational usually means selfish --- but not always• Rich history, flourished during the Cold War• Traditionally viewed as a subject of economics• Subsequently applied by many fields

– evolutionary biology, social psychology… now computer science

• Perhaps the branch of pure math most widely examined outside of the “hard” sciences

Prisoner’s Dilemma

• Cooperate = deny the crime; defect = confess guilt of both• Claim that (defect, defect) is an equilibrium:

– if I am definitely going to defect, you choose between -10 and -8– so you will also defect– same logic applies to me

• Note unilateral nature of equilibrium:– I fix a behavior or strategy for you, then choose my best response

• Claim: no other pair of strategies is an equilibrium• But we would have been so much better off cooperating…

cooperate defect

cooperate -1, -1 -10, -0.25

defect -0.25, -10 -8, -8

Penny Matching

• What are the equilibrium strategies now?• There are none!

– if I play heads then you will of course play tails– but that makes me want to play tails too– which in turn makes you want to play heads– etc. etc. etc.

• But what if we can each (privately) flip coins?– the strategy pair (1/2, 1/2) is an equilibrium

• Such randomized strategies are called mixed strategies

heads tails

heads 1, 0 0, 1

tails 0, 1 1, 0

The World According to Nash

• A mixed strategy for a player is a distribution on their available actions– e.g. 1/3 rock, 1/3 paper, 1/3 scissors

• Joint mixed strategy for N players: – a distribution for each player (possibly different)– assume everyone knows all the distributions – but the “coin flips” used to select from player i’s distribution known only to i

• “private randomness”• so only player I knows their actual choice of action• can people randomize? (more later)

• Joint mixed strategy is an equilibrium if:– for every player i, their distribution is a best response to all the others

• i.e. cannot get higher (average or expected) payoff by changing distribution • only consider unilateral deviations by each player!

– Nash 1950: every game has a mixed strategy equilibrium!– no matter how many rows and columns there are– in fact, no matter how many players there are

• Thus known as a Nash equilibrium• A major reason for Nash’s Nobel Prize in economics

Facts about Nash Equilibria

• While there is always at least one, there might be many– zero-sum games: all equilibria give the same payoffs to each

player– non zero-sum: different equilibria may give different payoffs!

• Equilibrium is a static notion– does not suggest how players might learn to play equilibrium– does not suggest how we might choose among multiple

equilibria• Nash equilibrium is a strictly competitive notion

– players cannot have “pre-play communication”– bargains, side payments, threats, collusions, etc. not allowed

• Computing Nash equilibria for large games is difficult

Behavioral Game Theory:A Brief But Important Digression

Supplementary slides courtesy of Colin Camerer, CalTech

Behavioral Game Theoryand Game Practice

• Game theory: how rational individuals should behave

• Who are these rational individuals?• BGT: looks at how people actually behave

– experiment by setting up real economic situations– account for people’s economic decisions– don’t break game theory when it works

• Fit a model to observations, not “rationality”

Beauty Contest Game

• N players choose numbers xi in [0,100]• Compute target (2/3)*( xi /N)• Closest to target wins $20

Beauty Contest Analysis

Some number of players try to guess a number that is 2/3 of the average guess.

The answer can’t be between 68 and 100 - no use guessing in that interval. It is dominated.

But if no one guesses in that interval, the answer won’t be greater than 44.

But if no one guesses more than 44, the answer won’t be greater than 29…

Everyone should guess 0! And good game theorists would…

But they’d lose…

Beauty contest results (Expansion, Financial Times, Spektrum)

0.00

0.05

0.10

0.15

0.20

numbers

rela

tive

fr

eq

ue

nci

es

22 50 10033

average 23.07

0

Iterated Dominance

People don’t instantly compute all the way to 0

The median subject uses 1 or 2 rounds of iteration (25, 35)

Guessing 0 on the first round (game theorist) is poor

Guessing 30 (behavioral game theory) is much better

But 30 isn’t a good guess the seventh time you play…

Ultimatum Game

• Proposer has $10• Offers x to Responder (keeps $10-x)• What should the Responder do?

– Self-interest: Take any x>0– Empirical: Reject x=$2 half the time

How People Ultimatum-Bargain

Thousands of games have been played in experiments…

• In different cultures around the world• With different stakes• With different mixes of men and women• By students of different majors• Etc. etc. etc.

Pretty much always, two things prove true:

1. Player 1 offers close to, but less than, half (40% or so)

2. Player 2 rejects low offers (20% or less)

Ultimatum offers across societies (mean shaded, mode is largest

circle…)

Behavioral Game Theory:Some Key Themes

• Bounded Rationality: Humans don’t have unlimited computational/reasoning capacity (Beauty Contest)

• Inequality Aversion: Humans often deviate from equilibrium towards “fairness” (Ultimatum)

• Mixed Strategies: Humans can generate “random” values within limits; better if paid.

Board Games and Game Theory• What does game theory say about richer games?

– tic-tac-toe, checkers, backgammon, go,…– these are all games of complete information with state– incomplete information: poker ( “Bayes Nash” equilibrium)

• Imagine an absurdly large “game matrix” for chess:– each row/column represents a complete strategy for playing– strategy = a mapping from every possible board configuration to the next

move for the player– number of rows or columns is huge --- but finite!

• Thus, a Nash equilibrium for chess exists!– it’s just completely infeasible to compute it

• July 2007: Checkers is solved (see Fourth Column)

Games on Networks• Matrix game “networks”• Vertices are the players• Keeping the normal (tabular) form

– is expensive (exponential in N)– misses the point

• Most strategic/economic settings have much more structure – asymmetry in connections– local and global structure– special properties of payoffs

• Two broad types of structure:– special structure of the network

• e.g. geographically local connections

– special payoff functions• e.g. financial markets

Case Study: Interdependent Security Games

on Networks

The Airline Security Problem• Imagine an expensive new bomb-screening technology

– large cost C to invest in new technology– cost of a mid-air explosion: L >> C

• There are two sources of explosion risk to an airline:– risk from directly checked baggage: new technology can reduce this– risk from transferred baggage: new technology does nothing– transferred baggage not re-screened (except for El Al airlines)

• This is a “game”…– each player (airline) must choose between I(nvesting) or N(ot)

• partial investment ~ mixed strategy– (negative) payoff to player (cost of action) depends on all others

• …on a network– the network of transfers between air carriers– not the complete graph– best thought of as a weighted network

Abstract Features of the Game

• Direct and indirect sources of risk• Investment reduces/eliminates direct risk only• Risk is of a catastrophic event (L >> C)

– can effectively occur only once• May only have incentive to invest if enough others do!• Note: much more involved network interaction than info

transmittal, message forwarding, search, etc.

The IDS Model[Kunreuther and Heal]

• Let x_i be the fraction of the investment C airline i makes• p_i: probability of explosion due to directly checked bag• S_i: probability of “catching” a bomb from someone else

– a straightforward function of all the “neighboring” airlines j– incorporates both their investment decision j (x_j) and their

probability or rate of transfer to airline i• Payoff structure (qualititative, can be made quantitative):

– increasing x_i reduces “effective direct risk” below p_i…– …but at increasing cost (x_i*C)..– …and does nothing to reduce effective indirect risk S_i, which can

only be reduced by the investments of others– network structure influences S_i

• Typical strategic incentives:– when your neighbors are under-investing, your incentive to invest is

low• basic problem: so much indirect risk already that you can’t help yourself

much– when your neighbors are all fully investing, your incentive to invest is

high• because your fate is in your own control now --- can reduce your only

remaining source of risk• What are the Nash equilibria?

– fully connected network with uniform transfer rates: full investment or no investment by all parties!

Other IDS Settings• Fire prevention

– catastrophic event: destruction of condo– investment decision: fire sprinkler in unit

• Corporate malfeasance (Arthur Anderson)– catastrophic event: bankruptcy– “investment” decision: risk management/ethics practice

• Computer security– catastrophic event: erasure of shared disk– investment decision: upgrade of anti-virus software

• Vaccination– catastrophic event: contraction of disease– investment decision: vaccination– incentives are reversed in this setting

IDS: One Last Try…

• Consider perspective of a player with only one neighbor• C: cost of new technology• L: cost of disaster• p: player 1 “direct” risk• q: player 1 “contagion” risk from player 2• x: player 1 investment fraction• y: player 2 investment fraction• Expected cost to player 1:

– xC + (1-x)pL + (1-(1-x)p)[(1-y)q]L• For any fixed value of y, a linear function of x

– negative slope: incentive to invest– positive slope: no incentive to invesgt

investment cost

prob. of direct disaster x cost

probability of no direct disaster

probability of contagion disaster

independent of player 2; no incentive to invest

independent of player 2; incentive to invest

dependent on player 2; incentive to invest

dependent on player 2; incentive can be inverted

An Experimental Study• Data:

– 35K N. American civilian flight itineraries reserved on 8/26/02– each indicates full itinerary: airports, carriers, flight numbers– assume all direct risk probabilities p_i are small and equal– carrier-to-carrier xfer rates used for risk xfer probabilities

• The simulation:– carrier i begins at investment level x_i = 0– at each time step, for every carrier i:– carrier i computes costs of full and no investment unilaterally– adjusts investment level x_i in direction of improvement (gradient)

• An investigation of learning to play an equilibrium (?)

Network Visualization

Airport to airport Carrier to carrier

least busy

most busy

level of investment

simulationtime

The Price of Anarchy is High

Tipping and Cascading

Necessary Conditions for Tipping